In Exercises 21–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities
Thirty-six percent of Americans think there is still a need for the practice of changing their clocks for Daylight Savings Time. You randomly select seven Americans. Find the probability that the number who say there is still a need for changing their clocks for Daylight Savings Time is (a) exactly four

Verified step by step guidance

1. Step 1: Identify the type of distribution to use. Since the problem involves a fixed number of trials (7 Americans), each with two possible outcomes (agree or disagree), and a constant probability of success (36% or 0.36), this is a binomial distribution problem.
2. Step 2: Write down the formula for the binomial probability distribution: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n' is the number of trials, 'k' is the number of successes, 'p' is the probability of success, and '1-p' is the probability of failure.
3. Step 3: Substitute the given values into the formula. Here, n = 7 (number of trials), k = 4 (number of successes), and p = 0.36 (probability of success). The formula becomes: P(X = 4) = (7 choose 4) * (0.36)^4 * (1-0.36)^(7-4).
4. Step 4: Calculate the binomial coefficient (7 choose 4), which is given by the formula: (n choose k) = n! / [k! * (n-k)!]. Substitute n = 7 and k = 4 to compute this value.
5. Step 5: Multiply the binomial coefficient by the probabilities raised to their respective powers. Specifically, compute (0.36)^4 and (1-0.36)^3, then multiply these values together with the binomial coefficient to find the probability P(X = 4).

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, it applies to the scenario of selecting seven Americans, where each individual can either support or oppose the need for changing clocks. The probability of exactly four supporters can be calculated using the binomial formula, which incorporates the number of trials, the number of successes, and the probability of success.

Probability Calculation

Probability calculation involves determining the likelihood of a specific outcome occurring within a defined set of possibilities. For the binomial distribution, this is done using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n' is the total number of trials, 'k' is the number of successes, and 'p' is the probability of success. Understanding how to apply this formula is crucial for solving the given problem.

Unusual Events

An event is considered unusual if its probability is significantly low, typically defined as less than 5%. In the context of the problem, after calculating the probability of exactly four Americans supporting the need for Daylight Savings Time, one must assess whether this probability falls below the threshold for unusual events. This evaluation helps in understanding the significance of the result in a broader context.